On optimal transport maps between $\frac{1}{d}$-concave densities

Abstract

In this paper we extend the scope of Caffarelli’s contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in ${\mathbb{R}}^d$. Our focus is on a broader category of densities, specifically those that are $\frac{1}{d}$-concave and can be represented as $V^{-d}$, where $V$ is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli’s theorem.